Section 4.2: Maximum and Minimum Values

1. Section 4.2: Maximum and Minimum Values Practice HW from Stewart Textbook (not to hand in) p. 276 # 1-5 odd, 15-29 odd, 35, 37, 39, 43

2. Extrema Let D be the domain of a function f. • A function f has and absolutemaximum (global maximum) at x = c if for all x in D ( is the largest y value for the graph of f on the domain D). 2. A function f has and absoluteminimum (global minimum) at x = c if for all x in D ( is the smallest y value for the graph of f on the domain D).

3. The absolute maximum and absolute minimum values are known as extremevalues.

4. Example 1: Determine the absolute maximum and minimum values for the following graphs.

6. Relative Extrema • A function f has a local maximum (relative maximum) at x = c if when x is near c ( f changes from increasing to decreasing) at the point . • A function f has a local minimum (relative minimum) at x = c if when x is near c (f changes from decreasing to increasing) at the point .

7. Example 2: Determine the local (relative) maximum and minimum points for the following graphs. Solution:

9. Note: Local maximum and local minimum points do not always give absolute maximum and minimum points.

10. Critical Numbers If a function f is defined at x = c (x = c is in the domain of f ), then x = c is a critical number (critical point) if or if is undefined.

13. Fact: If f has a relative minimum or a relative maximum at x = c, then x = c must be a critical number for the function f. Note: Before determining the critical numbers for a function, you should state the domain of the function first.

14. Example 3: Find the critical numbers of the function . Solution:

15. Example 4: Find the critical numbers of the function . Solution:

16. Note Having x = c be a critical number, that is, when or is undefined, does not guarantee that x = c produces a local maximum or local minimum for the function f.

17. Example 5: Demonstrate that the function has a critical number but no local maximum or minimum point. Solution:

19. The Extreme Value Theorem If a function f is continuous on a closed interval [a, b], then f has both an absolute minimum and an absolute maximum in [a, b].

20. Steps for Evaluation Absolute Extrema on a Closed Interval To find the absolute maximum and absolute minimum points for a continuous function f on the closed interval [a, b]. • Find the critical numbers of f (values of x where or is undefined) that are contained in [a, b]. Important! You must make sure you only consider critical numbers for step 2 that are in [a, b]. For critical numbers not in [a, b], you must throw these out and not consider them for step 2. 2. Evaluate f (find the y values) at each critical number in [a, b] and at the endpoints of the interval x = a and x = b.

21. 3. The smallest of these values (smallest y value) from step 2 is the absolute minimum. The largest of these values (largest y value) is the absolute maximum.

22. Example 6: Find the absolute maximum and absolute minimum values for the function on the interval [0, 3]. Solution:

25. Example 7: Find the absolute maximum and absolute minimum values for the function on the interval [1, 4]. Solution:

28. Example 8: Find the absolute maximum and absolute minimum values for the function on the interval . Solution: (In typewritten notes)